QUESTION IMAGE
Question
the iq scores of 50 students are given below.
112 115 119 86 111 100 116 97 113 96
105 116 95 116 105 89 90 89 100 94
91 115 98 90 118 113 119 85 114 111
113 94 102 88 85 108 98 102 108 118
100 109 111 107 90 119 85 112 92 93
(a) construct a grouped frequency distribution for the data. use 85-89 for the first class and use the same width for each subsequent class.
class frequency
85 - 89
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(b) which of the following is the correct histogram for this data?
Part (a)
Step 1: Determine Class Width
The first class is 85 - 89, so the class width is \( 89 - 85 + 1 = 5 \) (we add 1 because the class includes both endpoints). Wait, actually, for continuous data (IQ scores can be considered as continuous for grouping), the class width is \( 89 - 85 = 4 \)? Wait, no, let's check the data. Wait, 85, 86, 88, 89, 85, 85, 88. Wait, the first class is 85 - 89. Let's list all data points:
First, list all 50 IQ scores:
112, 115, 119, 86, 111, 100, 116, 97, 113, 96,
105, 116, 95, 116, 105, 89, 90, 89, 100, 94,
91, 115, 98, 90, 118, 113, 119, 85, 114, 111,
113, 94, 102, 88, 85, 108, 98, 102, 108, 118,
100, 109, 111, 107, 90, 119, 85, 112, 92, 93.
Now, first class: 85 - 89. Let's count how many are in 85 - 89:
85, 86, 89, 89, 85, 88, 85. Let's count: 85 (3 times), 86 (1), 88 (1), 89 (2). Total: 3 + 1 + 1 + 2 = 7? Wait, let's list them:
From the data:
First row: 86
Second row: 89, 89
Third row: 85
Fourth row: 88, 85
Fifth row: 85
So 85 (3), 86 (1), 88 (1), 89 (2). Total: 3 + 1 + 1 + 2 = 7. So frequency for 85 - 89 is 7.
Next class: 90 - 94 (since class width is 5? Wait, 85 - 89, next would be 90 - 94 (89 + 1 = 90, 94 = 90 + 4? Wait, no, 85 - 89: 85,86,87,88,89 (5 numbers). So class width is 5 (inclusive). So next class: 90 - 94 (90,91,92,93,94). Let's count:
Data points in 90 - 94:
Second row: 90, 94
Third row: 91
Fourth row: 94
Fifth row: 90, 92, 93
Let's list:
90 (2), 91 (1), 92 (1), 93 (1), 94 (2). Total: 2 + 1 + 1 + 1 + 2 = 7? Wait:
Second row: 90, 94 → 2
Third row: 91 → 1 (total 3)
Fourth row: 94 → 1 (total 4)
Fifth row: 90, 92, 93 → 3 (total 7). Yes, 7.
Next class: 95 - 99 (95,96,97,98,99). Data points:
First row: 97, 96
Second row: 95
Third row: 98
Fourth row: 98
So 95 (1), 96 (1), 97 (1), 98 (2). Total: 1 + 1 + 1 + 2 = 5.
Next class: 100 - 104 (100,101,102,103,104). Data points:
First row: 100
Second row: 100
Fourth row: 102, 102
So 100 (2), 102 (2). Total: 4.
Next class: 105 - 109 (105,106,107,108,109). Data points:
Second row: 105, 105
Fourth row: 108, 108
Fifth row: 109, 107
So 105 (2), 107 (1), 108 (2), 109 (1). Total: 2 + 1 + 2 + 1 = 6.
Next class: 110 - 114 (110,111,112,113,114). Data points:
First row: 112, 111, 113
Second row: (none)
Third row: 114, 111
Fourth row: 113
Fifth row: 111, 112
So 111 (3), 112 (2), 113 (2), 114 (1). Total: 3 + 2 + 2 + 1 = 8.
Next class: 115 - 119 (115,116,117,118,119). Data points:
First row: 115, 119, 116
Second row: 116, 116
Third row: 115, 118, 119
Fourth row: 118
Fifth row: 119
So 115 (2), 116 (3), 118 (2), 119 (3). Total: 2 + 3 + 2 + 3 = 10.
Wait, let's check total frequency: 7 (85-89) + 7 (90-94) + 5 (95-99) + 4 (100-104) + 6 (105-109) + 8 (110-114) + 10 (115-119) = 7+7=14, +5=19, +4=23, +6=29, +8=37, +10=47. Wait, that's 47, but we have 50 data points. Did I miss some?
Wait, let's recount each class:
85-89:
Data points: 85 (3), 86 (1), 88 (1), 89 (2). Wait, 3+1+1+2=7. Correct.
90-94:
90 (2), 91 (1), 92 (1), 93 (1), 94 (2). 2+1+1+1+2=7. Correct.
95-99:
95 (1), 96 (1), 97 (1), 98 (2). 1+1+1+2=5. Correct.
100-104:
100 (2), 102 (2). Any others? 101? No. So 4. Correct.
105-109:
105 (2), 107 (1), 108 (2), 109 (1). Any 106? No. So 2+1+2+1=6. Correct.
110-114:
111 (3), 112 (2), 113 (2), 114 (1). Any 110? No. So 3+2+2+1=8. Correct.
115-119:
115 (2), 116 (3), 118 (2), 119 (3). Any 117? No. So 2+3+2+3=10. Correct.
Wait, 7+7=14, +5=19, +4=23, +6=29, +8=37, +10=47. Missing 3. Oh, maybe I made a mistake in class width. Wait, maybe the class width is 5, but inclusive of the upper limit? Wait, maybe the f…
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Step 1: Determine Class Width
The first class is 85 - 89, so the class width is \( 89 - 85 + 1 = 5 \) (we add 1 because the class includes both endpoints). Wait, actually, for continuous data (IQ scores can be considered as continuous for grouping), the class width is \( 89 - 85 = 4 \)? Wait, no, let's check the data. Wait, 85, 86, 88, 89, 85, 85, 88. Wait, the first class is 85 - 89. Let's list all data points:
First, list all 50 IQ scores:
112, 115, 119, 86, 111, 100, 116, 97, 113, 96,
105, 116, 95, 116, 105, 89, 90, 89, 100, 94,
91, 115, 98, 90, 118, 113, 119, 85, 114, 111,
113, 94, 102, 88, 85, 108, 98, 102, 108, 118,
100, 109, 111, 107, 90, 119, 85, 112, 92, 93.
Now, first class: 85 - 89. Let's count how many are in 85 - 89:
85, 86, 89, 89, 85, 88, 85. Let's count: 85 (3 times), 86 (1), 88 (1), 89 (2). Total: 3 + 1 + 1 + 2 = 7? Wait, let's list them:
From the data:
First row: 86
Second row: 89, 89
Third row: 85
Fourth row: 88, 85
Fifth row: 85
So 85 (3), 86 (1), 88 (1), 89 (2). Total: 3 + 1 + 1 + 2 = 7. So frequency for 85 - 89 is 7.
Next class: 90 - 94 (since class width is 5? Wait, 85 - 89, next would be 90 - 94 (89 + 1 = 90, 94 = 90 + 4? Wait, no, 85 - 89: 85,86,87,88,89 (5 numbers). So class width is 5 (inclusive). So next class: 90 - 94 (90,91,92,93,94). Let's count:
Data points in 90 - 94:
Second row: 90, 94
Third row: 91
Fourth row: 94
Fifth row: 90, 92, 93
Let's list:
90 (2), 91 (1), 92 (1), 93 (1), 94 (2). Total: 2 + 1 + 1 + 1 + 2 = 7? Wait:
Second row: 90, 94 → 2
Third row: 91 → 1 (total 3)
Fourth row: 94 → 1 (total 4)
Fifth row: 90, 92, 93 → 3 (total 7). Yes, 7.
Next class: 95 - 99 (95,96,97,98,99). Data points:
First row: 97, 96
Second row: 95
Third row: 98
Fourth row: 98
So 95 (1), 96 (1), 97 (1), 98 (2). Total: 1 + 1 + 1 + 2 = 5.
Next class: 100 - 104 (100,101,102,103,104). Data points:
First row: 100
Second row: 100
Fourth row: 102, 102
So 100 (2), 102 (2). Total: 4.
Next class: 105 - 109 (105,106,107,108,109). Data points:
Second row: 105, 105
Fourth row: 108, 108
Fifth row: 109, 107
So 105 (2), 107 (1), 108 (2), 109 (1). Total: 2 + 1 + 2 + 1 = 6.
Next class: 110 - 114 (110,111,112,113,114). Data points:
First row: 112, 111, 113
Second row: (none)
Third row: 114, 111
Fourth row: 113
Fifth row: 111, 112
So 111 (3), 112 (2), 113 (2), 114 (1). Total: 3 + 2 + 2 + 1 = 8.
Next class: 115 - 119 (115,116,117,118,119). Data points:
First row: 115, 119, 116
Second row: 116, 116
Third row: 115, 118, 119
Fourth row: 118
Fifth row: 119
So 115 (2), 116 (3), 118 (2), 119 (3). Total: 2 + 3 + 2 + 3 = 10.
Wait, let's check total frequency: 7 (85-89) + 7 (90-94) + 5 (95-99) + 4 (100-104) + 6 (105-109) + 8 (110-114) + 10 (115-119) = 7+7=14, +5=19, +4=23, +6=29, +8=37, +10=47. Wait, that's 47, but we have 50 data points. Did I miss some?
Wait, let's recount each class:
85-89:
Data points: 85 (3), 86 (1), 88 (1), 89 (2). Wait, 3+1+1+2=7. Correct.
90-94:
90 (2), 91 (1), 92 (1), 93 (1), 94 (2). 2+1+1+1+2=7. Correct.
95-99:
95 (1), 96 (1), 97 (1), 98 (2). 1+1+1+2=5. Correct.
100-104:
100 (2), 102 (2). Any others? 101? No. So 4. Correct.
105-109:
105 (2), 107 (1), 108 (2), 109 (1). Any 106? No. So 2+1+2+1=6. Correct.
110-114:
111 (3), 112 (2), 113 (2), 114 (1). Any 110? No. So 3+2+2+1=8. Correct.
115-119:
115 (2), 116 (3), 118 (2), 119 (3). Any 117? No. So 2+3+2+3=10. Correct.
Wait, 7+7=14, +5=19, +4=23, +6=29, +8=37, +10=47. Missing 3. Oh, maybe I made a mistake in class width. Wait, maybe the class width is 5, but inclusive of the upper limit? Wait, maybe the first class is 85-89 (width 5), next 90-94 (width 5), then 95-99, 100-104, 105-109, 110-114, 115-119. Wait, let's list all data points again and count:
List of all 50 scores:
- 112
- 115
- 119
- 86
- 111
- 100
- 116
- 97
- 113
- 96
- 105
- 116
- 95
- 116
- 105
- 89
- 90
- 89
- 100
- 94
- 91
- 115
- 98
- 90
- 118
- 113
- 119
- 85
- 114
- 111
- 113
- 94
- 102
- 88
- 85
- 108
- 98
- 102
- 108
- 118
- 100
- 109
- 111
- 107
- 90
- 119
- 85
- 112
- 92
- 93
Now, let's count each class:
85-89 (inclusive, so 85 ≤ x ≤ 89):
Scores: 4 (86), 16 (89), 18 (89), 28 (85), 34 (88), 35 (85), 47 (85). So that's 7 scores (4,16,18,28,34,35,47). Correct, 7.
90-94 (90 ≤ x ≤ 94):
17 (90), 20 (94), 21 (91), 24 (90), 32 (94), 45 (90), 49 (92), 50 (93). Wait, 17,20,21,24,32,45,49,50. That's 8 scores! Oh, I missed 49 (92) and 50 (93) earlier. So 8, not 7. Let's recount:
17:90, 20:94, 21:91, 24:90, 32:94, 45:90, 49:92, 50:93. So 8 scores.
95-99 (95 ≤ x ≤ 99):
8 (97), 10 (96), 13 (95), 23 (98), 37 (98). So 5 scores (8,10,13,23,37). Correct, 5.
100-104 (100 ≤ x ≤ 104):
6 (100), 19 (100), 33 (102), 38 (102), 41 (100). Wait, 6,19,33,38,41. That's 5 scores! Oh, I missed 41 (100) earlier. So 5, not 4.
105-109 (105 ≤ x ≤ 109):
11 (105), 15 (105), 36 (108), 39 (108), 42 (109), 44 (107). So 6 scores (11,15,36,39,42,44). Correct, 6.
110-114 (110 ≤ x ≤ 114):
1 (112), 5 (111), 9 (113), 29 (114), 30 (111), 31 (113), 43 (111), 48 (112). So 8 scores (1,5,9,29,30,31,43,48). Correct, 8.
115-119 (115 ≤ x ≤ 119):
2 (115), 3 (119), 7 (116), 12 (116), 14 (116), 22 (115), 25 (118), 27 (119), 40 (118), 46 (119). So 10 scores (2,3,7,12,14,22,25,27,40,46). Correct, 10.
Now total: 7 (85-89) + 8 (90-94) + 5 (95-99) + 5 (100-104) + 6 (105-109) + 8 (110-114) + 10 (115-119) = 7+8=15, +5=20, +5=25, +6=31, +8=39, +10=49. Wait, still missing 1. Oh no, where's the 50th? Wait, let's count the list: 1-50, so 50 scores. Let's check again:
85-89: 7 (4,16,18,28,34,35,47) → 7
90-94: 8 (17,20,21,24,32,45,49,50) → 8 (total 15)
95-99: 5 (8,10,13,23,37) → 5 (total 20)
100-104: 5 (6,19,33,38,41) → 5 (total 25)
105-109: 6 (11,15,36,39,42,44) → 6 (total 31)
110-114: 8 (1,5,9,29,30,31,43,48) → 8 (total 39)
115-119: 10 (2,3,7,12,14,22,25,27,40,46) → 10 (total 49). Wait, missing score 4? No, score 4 is 86 (in 85-89). Wait, maybe I miscounted 100-104. Let's check 100-104: 6 (100), 19 (100), 33 (102), 38 (102), 41 (100). That's 5. Any other? 101? No. Wait, score 41 is 100, yes. So 5. Then where's the 50th? Oh! Wait, score 23 is 98 (in 95-99), score 37 is 98 (in 95-99). Wait, score 23: 98, score 37: 98. So 95-99 has 5: 8 (97